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In mathematics, a group is a
set Set, The Set, SET or SETS may refer to: Science, technology, and mathematics Mathematics *Set (mathematics), a collection of elements *Category of sets, the category whose objects and morphisms are sets and total functions, respectively Electro ...
and an
operation Operation or Operations may refer to: Arts, entertainment and media * ''Operation'' (game), a battery-operated board game that challenges dexterity * Operation (music), a term used in musical set theory * ''Operations'' (magazine), Multi-Ma ...
that combines any two elements of the set to produce a third element of the set, in such a way that the operation is
associative In mathematics, the associative property is a property of some binary operations, which means that rearranging the parentheses in an expression will not change the result. In propositional logic, associativity is a valid rule of replacement ...
, an identity element exists and every element has an inverse. These three
axiom An axiom, postulate, or assumption is a statement that is taken to be true, to serve as a premise or starting point for further reasoning and arguments. The word comes from the Ancient Greek word (), meaning 'that which is thought worthy or f ...
s hold for
number systems A number is a mathematical object used to count, measure, and label. The original examples are the natural numbers 1, 2, 3, 4, and so forth. Numbers can be represented in language with number words. More universally, individual numbers ...
and many other mathematical structures. For example, the
integer An integer is the number zero (), a positive natural number (, , , etc.) or a negative integer with a minus sign (−1, −2, −3, etc.). The negative numbers are the additive inverses of the corresponding positive numbers. In the languag ...
s together with the addition operation form a group. The concept of a group and the axioms that define it were elaborated for handling, in a unified way, essential structural properties of very different mathematical entities such as numbers, geometric shapes and polynomial roots. Because the concept of groups is ubiquitous in numerous areas both within and outside mathematics, some authors consider it as a central organizing principle of contemporary mathematics. In
geometry Geometry (; ) is, with arithmetic, one of the oldest branches of mathematics. It is concerned with properties of space such as the distance, shape, size, and relative position of figures. A mathematician who works in the field of geometry is ca ...
groups arise naturally in the study of
symmetries Symmetry (from grc, συμμετρία "agreement in dimensions, due proportion, arrangement") in everyday language refers to a sense of harmonious and beautiful proportion and balance. In mathematics, "symmetry" has a more precise definiti ...
and
geometric transformation In mathematics, a geometric transformation is any bijection of a set to itself (or to another such set) with some salient geometrical underpinning. More specifically, it is a function whose domain and range are sets of points — most often b ...
s: The symmetries of an object form a group, called the
symmetry group In group theory, the symmetry group of a geometric object is the group of all transformations under which the object is invariant, endowed with the group operation of composition. Such a transformation is an invertible mapping of the am ...
of the object, and the transformations of a given type form a general group. Lie groups appear in symmetry groups in geometry, and also in the
Standard Model The Standard Model of particle physics is the theory describing three of the four known fundamental forces (electromagnetic, weak and strong interactions - excluding gravity) in the universe and classifying all known elementary particles. It ...
of
particle physics Particle physics or high energy physics is the study of fundamental particles and forces that constitute matter and radiation. The fundamental particles in the universe are classified in the Standard Model as fermions (matter particles) an ...
. The Poincaré group is a Lie group consisting of the symmetries of
spacetime In physics, spacetime is a mathematical model that combines the three dimensions of space and one dimension of time into a single four-dimensional manifold. Spacetime diagrams can be used to visualize relativistic effects, such as why differe ...
in
special relativity In physics, the special theory of relativity, or special relativity for short, is a scientific theory regarding the relationship between space and time. In Albert Einstein's original treatment, the theory is based on two postulates: # The laws ...
. Point groups describe symmetry in molecular chemistry. The concept of a group arose in the study of polynomial equations, starting with
Évariste Galois Évariste Galois (; ; 25 October 1811 – 31 May 1832) was a French mathematician and political activist. While still in his teens, he was able to determine a necessary and sufficient condition for a polynomial to be solvable by radicals ...
in the 1830s, who introduced the term ''group'' (French: ) for the symmetry group of the
roots A root is the part of a plant, generally underground, that anchors the plant body, and absorbs and stores water and nutrients. Root or roots may also refer to: Art, entertainment, and media * ''The Root'' (magazine), an online magazine focusing ...
of an equation, now called a Galois group. After contributions from other fields such as number theory and geometry, the group notion was generalized and firmly established around 1870. Modern
group theory In abstract algebra, group theory studies the algebraic structures known as groups. The concept of a group is central to abstract algebra: other well-known algebraic structures, such as rings, fields, and vector spaces, can all be seen as ...
—an active mathematical discipline—studies groups in their own right. To explore groups, mathematicians have devised various notions to break groups into smaller, better-understandable pieces, such as
subgroup In group theory, a branch of mathematics, given a group ''G'' under a binary operation ∗, a subset ''H'' of ''G'' is called a subgroup of ''G'' if ''H'' also forms a group under the operation ∗. More precisely, ''H'' is a subgro ...
s, quotient groups and
simple group SIMPLE Group Limited is a conglomeration of separately run companies that each has its core area in International Consulting. The core business areas are Legal Services, Fiduciary Activities, Banking Intermediation and Corporate Service. The d ...
s. In addition to their abstract properties, group theorists also study the different ways in which a group can be expressed concretely, both from a point of view of
representation theory Representation theory is a branch of mathematics that studies abstract algebraic structures by ''representing'' their elements as linear transformations of vector spaces, and studies modules over these abstract algebraic structures. In esse ...
(that is, through the representations of the group) and of computational group theory. A theory has been developed for finite groups, which culminated with the classification of finite simple groups, completed in 2004. Since the mid-1980s,
geometric group theory Geometric group theory is an area in mathematics devoted to the study of finitely generated groups via exploring the connections between algebraic properties of such groups and topological and geometric properties of spaces on which these groups ...
, which studies finitely generated groups as geometric objects, has become an active area in group theory.

Definition and illustration

First example: the integers

One of the more familiar groups is the set of
integer An integer is the number zero (), a positive natural number (, , , etc.) or a negative integer with a minus sign (−1, −2, −3, etc.). The negative numbers are the additive inverses of the corresponding positive numbers. In the languag ...
s $\Z = \$ together with
addition Addition (usually signified by the plus symbol ) is one of the four basic operations of arithmetic, the other three being subtraction, multiplication and division. The addition of two whole numbers results in the total amount or '' sum'' of ...
. For any two integers $a$ and $b$, the sum $a+b$ is also an integer; this '' closure'' property says that $+$ is a
binary operation In mathematics, a binary operation or dyadic operation is a rule for combining two elements (called operands) to produce another element. More formally, a binary operation is an operation of arity two. More specifically, an internal binary op ...
on $\Z$. The following properties of integer addition serve as a model for the group axioms in the definition below. *For all integers $a$, $b$ and $c$, one has $\left(a+b\right)+c=a+\left(b+c\right)$. Expressed in words, adding $a$ to $b$ first, and then adding the result to $c$ gives the same final result as adding $a$ to the sum of $b$ and $c$. This property is known as ''
associativity In mathematics, the associative property is a property of some binary operations, which means that rearranging the parentheses in an expression will not change the result. In propositional logic, associativity is a valid rule of replacemen ...
''. *If $a$ is any integer, then $0+a=a$ and $a+0=a$.
Zero 0 (zero) is a number representing an empty quantity. In place-value notation such as the Hindu–Arabic numeral system, 0 also serves as a placeholder numerical digit, which works by multiplying digits to the left of 0 by the radix, usu ...
is called the '' identity element'' of addition because adding it to any integer returns the same integer. *For every integer $a$, there is an integer $b$ such that $a+b=0$ and $b+a=0$. The integer $b$ is called the '' inverse element'' of the integer $a$ and is denoted $-a$. The integers, together with the operation $+$, form a mathematical object belonging to a broad class sharing similar structural aspects. To appropriately understand these structures as a collective, the following definition is developed.

Definition

A group is a
set Set, The Set, SET or SETS may refer to: Science, technology, and mathematics Mathematics *Set (mathematics), a collection of elements *Category of sets, the category whose objects and morphisms are sets and total functions, respectively Electro ...
$G$ together with a
binary operation In mathematics, a binary operation or dyadic operation is a rule for combining two elements (called operands) to produce another element. More formally, a binary operation is an operation of arity two. More specifically, an internal binary op ...
on $G$, here denoted "$\cdot$", that combines any two elements $a$ and $b$ to form an element of $G$, denoted $a\cdot b$, such that the following three requirements, known as ''group axioms'', are satisfied: ;Associativity: For all $a$, $b$, $c$ in $G$, one has $\left(a\cdot b\right)\cdot c=a\cdot\left(b\cdot c\right)$. ;Identity element: There exists an element $e$ in $G$ such that, for every $a$ in $G$, one has $e\cdot a=a$ and $a\cdot e=a$. :Such an element is unique ( see below). It is called ''the identity element'' of the group. ;Inverse element: For each $a$ in $G$, there exists an element $b$ in $G$ such that $a\cdot b=e$ and $b\cdot a=e$, where $e$ is the identity element. :For each $a$, the element $b$ is unique ( see below); it is called ''the inverse'' of $a$ and is commonly denoted $a^$.

Notation and terminology

Formally, the group is the
ordered pair In mathematics, an ordered pair (''a'', ''b'') is a pair of objects. The order in which the objects appear in the pair is significant: the ordered pair (''a'', ''b'') is different from the ordered pair (''b'', ''a'') unless ''a'' = ''b''. (In con ...
of a set and a binary operation on this set that satisfies the group axioms. The set is called the ''underlying set'' of the group, and the operation is called the ''group operation'' or the ''group law''. A group and its underlying set are thus two different
mathematical object A mathematical object is an abstract concept arising in mathematics. In the usual language of mathematics, an ''object'' is anything that has been (or could be) formally defined, and with which one may do deductive reasoning and mathematical ...
s. To avoid cumbersome notation, it is common to abuse notation by using the same symbol to denote both. This reflects also an informal way of thinking: that the group is the same as the set except that it has been enriched by additional structure provided by the operation. For example, consider the set of real numbers $\R$, which has the operations of addition $a+b$ and
multiplication Multiplication (often denoted by the cross symbol , by the mid-line dot operator , by juxtaposition, or, on computers, by an asterisk ) is one of the four elementary mathematical operations of arithmetic, with the other ones being addi ...
$ab$. Formally, $\R$ is a set, $\left(\R,+\right)$ is a group, and $\left(\R,+,\cdot\right)$ is a field. But it is common to write $\R$ to denote any of these three objects. The ''additive group'' of the field $\R$ is the group whose underlying set is $\R$ and whose operation is addition. The ''multiplicative group'' of the field $\R$ is the group $\R^$ whose underlying set is the set of nonzero real numbers $\R \smallsetminus \$ and whose operation is multiplication. More generally, one speaks of an ''additive group'' whenever the group operation is notated as addition; in this case, the identity is typically denoted $0$, and the inverse of an element $x$ is denoted $-x$. Similarly, one speaks of a ''multiplicative group'' whenever the group operation is notated as multiplication; in this case, the identity is typically denoted $1$, and the inverse of an element $x$ is denoted $x^$. In a multiplicative group, the operation symbol is usually omitted entirely, so that the operation is denoted by juxtaposition, $ab$ instead of $a\cdot b$. The definition of a group does not require that $a\cdot b=b\cdot a$ for all elements $a$ and $b$ in $G$. If this additional condition holds, then the operation is said to be commutative, and the group is called an
abelian group In mathematics, an abelian group, also called a commutative group, is a group in which the result of applying the group operation to two group elements does not depend on the order in which they are written. That is, the group operation is commu ...
. It is a common convention that for an abelian group either additive or multiplicative notation may be used, but for a nonabelian group only multiplicative notation is used. Several other notations are commonly used for groups whose elements are not numbers. For a group whose elements are functions, the operation is often
function composition In mathematics, function composition is an operation that takes two functions and , and produces a function such that . In this operation, the function is applied to the result of applying the function to . That is, the functions and ...
$f\circ g$; then the identity may be denoted id. In the more specific cases of
geometric transformation In mathematics, a geometric transformation is any bijection of a set to itself (or to another such set) with some salient geometrical underpinning. More specifically, it is a function whose domain and range are sets of points — most often b ...
groups,
symmetry Symmetry (from grc, συμμετρία "agreement in dimensions, due proportion, arrangement") in everyday language refers to a sense of harmonious and beautiful proportion and balance. In mathematics, "symmetry" has a more precise definit ...
groups,
permutation group In mathematics, a permutation group is a group ''G'' whose elements are permutations of a given set ''M'' and whose group operation is the composition of permutations in ''G'' (which are thought of as bijective functions from the set ''M'' to ...
s, and automorphism groups, the symbol $\circ$ is often omitted, as for multiplicative groups. Many other variants of notation may be encountered.

Second example: a symmetry group

Two figures in the plane are congruent if one can be changed into the other using a combination of
rotation Rotation, or spin, is the circular movement of an object around a '' central axis''. A two-dimensional rotating object has only one possible central axis and can rotate in either a clockwise or counterclockwise direction. A three-dimensional ...
s, reflections, and
translation Translation is the communication of the meaning of a source-language text by means of an equivalent target-language text. The English language draws a terminological distinction (which does not exist in every language) between ''translat ...
s. Any figure is congruent to itself. However, some figures are congruent to themselves in more than one way, and these extra congruences are called
symmetries Symmetry (from grc, συμμετρία "agreement in dimensions, due proportion, arrangement") in everyday language refers to a sense of harmonious and beautiful proportion and balance. In mathematics, "symmetry" has a more precise definiti ...
. A
square In Euclidean geometry, a square is a regular quadrilateral, which means that it has four equal sides and four equal angles (90- degree angles, π/2 radian angles, or right angles). It can also be defined as a rectangle with two equal-lengt ...
has eight symmetries. These are: * the identity operation leaving everything unchanged, denoted id; * rotations of the square around its center by 90°, 180°, and 270° clockwise, denoted by $r_1$, $r_2$ and $r_3$, respectively; * reflections about the horizontal and vertical middle line ($f_$ and $f_$), or through the two
diagonal In geometry, a diagonal is a line segment joining two vertices of a polygon or polyhedron, when those vertices are not on the same edge. Informally, any sloping line is called diagonal. The word ''diagonal'' derives from the ancient Greek δ ...
s ($f_$ and $f_$). These symmetries are functions. Each sends a point in the square to the corresponding point under the symmetry. For example, $r_1$ sends a point to its rotation 90° clockwise around the square's center, and $f_$ sends a point to its reflection across the square's vertical middle line. Composing two of these symmetries gives another symmetry. These symmetries determine a group called the
dihedral group In mathematics, a dihedral group is the group of symmetries of a regular polygon, which includes rotations and reflections. Dihedral groups are among the simplest examples of finite groups, and they play an important role in group theory, ...
of degree four, denoted $\mathrm_4$. The underlying set of the group is the above set of symmetries, and the group operation is function composition. Two symmetries are combined by composing them as functions, that is, applying the first one to the square, and the second one to the result of the first application. The result of performing first $a$ and then $b$ is written symbolically ''from right to left'' as $b\circ a$ ("apply the symmetry $b$ after performing the symmetry $a$"). This is the usual notation for composition of functions. The group table lists the results of all such compositions possible. For example, rotating by 270° clockwise ($r_3$) and then reflecting horizontally ($f_$) is the same as performing a reflection along the diagonal ($f_$). Using the above symbols, highlighted in blue in the group table: $f_\mathrm h \circ r_3= f_\mathrm d.$ Given this set of symmetries and the described operation, the group axioms can be understood as follows. ''Binary operation'': Composition is a binary operation. That is, $a\circ b$ is a symmetry for any two symmetries $a$ and $b$. For example, $r_3\circ f_\mathrm h = f_\mathrm c,$ that is, rotating 270° clockwise after reflecting horizontally equals reflecting along the counter-diagonal ($f_$). Indeed, every other combination of two symmetries still gives a symmetry, as can be checked using the group table. ''Associativity'': The associativity axiom deals with composing more than two symmetries: Starting with three elements $a$, $b$ and $c$ of $\mathrm_4$, there are two possible ways of using these three symmetries in this order to determine a symmetry of the square. One of these ways is to first compose $a$ and $b$ into a single symmetry, then to compose that symmetry with $c$. The other way is to first compose $b$ and $c$, then to compose the resulting symmetry with $a$. These two ways must give always the same result, that is, $(a\circ b)\circ c = a\circ (b\circ c),$ For example, $\left(f_\circ f_\right)\circ r_2=f_\circ \left(f_\circ r_2\right)$ can be checked using the group table: $\begin (f_\mathrm d\circ f_\mathrm v)\circ r_2 &=r_3\circ r_2=r_1\\ f_\mathrm d\circ (f_\mathrm v\circ r_2) &=f_\mathrm d\circ f_\mathrm h =r_1. \end$ ''Identity element'': The identity element is $\mathrm$, as it does not change any symmetry $a$ when composed with it either on the left or on the right. ''Inverse element'': Each symmetry has an inverse: $\mathrm$, the reflections $f_$, $f_$, $f_$, $f_$ and the 180° rotation $r_2$ are their own inverse, because performing them twice brings the square back to its original orientation. The rotations $r_3$ and $r_1$ are each other's inverses, because rotating 90° and then rotation 270° (or vice versa) yields a rotation over 360° which leaves the square unchanged. This is easily verified on the table. In contrast to the group of integers above, where the order of the operation is immaterial, it does matter in $\mathrm_4$, as, for example, $f_\circ r_1=f_$ but $r_1\circ f_=f_$. In other words, $\mathrm_4$ is not abelian.

History

The modern concept of an abstract group developed out of several fields of mathematics. The original motivation for group theory was the quest for solutions of polynomial equations of degree higher than 4. The 19th-century French mathematician
Évariste Galois Évariste Galois (; ; 25 October 1811 – 31 May 1832) was a French mathematician and political activist. While still in his teens, he was able to determine a necessary and sufficient condition for a polynomial to be solvable by radicals ...
, extending prior work of Paolo Ruffini and Joseph-Louis Lagrange, gave a criterion for the solvability of a particular polynomial equation in terms of the
symmetry group In group theory, the symmetry group of a geometric object is the group of all transformations under which the object is invariant, endowed with the group operation of composition. Such a transformation is an invertible mapping of the am ...
of its
roots A root is the part of a plant, generally underground, that anchors the plant body, and absorbs and stores water and nutrients. Root or roots may also refer to: Art, entertainment, and media * ''The Root'' (magazine), an online magazine focusing ...
(solutions). The elements of such a Galois group correspond to certain permutations of the roots. At first, Galois's ideas were rejected by his contemporaries, and published only posthumously. More general permutation groups were investigated in particular by
Augustin Louis Cauchy Baron Augustin-Louis Cauchy (, ; ; 21 August 178923 May 1857) was a French mathematician, engineer, and physicist who made pioneering contributions to several branches of mathematics, including mathematical analysis and continuum mechanics. He w ...
.
Arthur Cayley Arthur Cayley (; 16 August 1821 – 26 January 1895) was a prolific British mathematician who worked mostly on algebra. He helped found the modern British school of pure mathematics. As a child, Cayley enjoyed solving complex maths problems ...
's ''On the theory of groups, as depending on the symbolic equation $\theta^n=1$'' (1854) gives the first abstract definition of a finite group. Geometry was a second field in which groups were used systematically, especially symmetry groups as part of Felix Klein's 1872 Erlangen program. After novel geometries such as hyperbolic and
projective geometry In mathematics, projective geometry is the study of geometric properties that are invariant with respect to projective transformations. This means that, compared to elementary Euclidean geometry, projective geometry has a different setting, pr ...
had emerged, Klein used group theory to organize them in a more coherent way. Further advancing these ideas, Sophus Lie founded the study of Lie groups in 1884. The third field contributing to group theory was number theory. Certain abelian group structures had been used implicitly in
Carl Friedrich Gauss Johann Carl Friedrich Gauss (; german: Gauß ; la, Carolus Fridericus Gauss; 30 April 177723 February 1855) was a German mathematician and physicist who made significant contributions to many fields in mathematics and science. Sometimes refer ...
's number-theoretical work '' Disquisitiones Arithmeticae'' (1798), and more explicitly by
Leopold Kronecker Leopold Kronecker (; 7 December 1823 – 29 December 1891) was a German mathematician who worked on number theory, algebra and logic. He criticized Georg Cantor's work on set theory, and was quoted by as having said, "'" ("God made the integers ...
. In 1847,
Ernst Kummer Ernst Eduard Kummer (29 January 1810 – 14 May 1893) was a German mathematician. Skilled in applied mathematics, Kummer trained German army officers in ballistics; afterwards, he taught for 10 years in a '' gymnasium'', the German equivalent of h ...
made early attempts to prove Fermat's Last Theorem by developing groups describing factorization into prime numbers. The convergence of these various sources into a uniform theory of groups started with Camille Jordan's (1870).
Walther von Dyck Walther Franz Anton von Dyck (6 December 1856 – 5 November 1934), born Dyck () and later ennobled, was a German mathematician. He is credited with being the first to define a mathematical group, in the modern sense in . He laid the foundations ...
(1882) introduced the idea of specifying a group by means of generators and relations, and was also the first to give an axiomatic definition of an "abstract group", in the terminology of the time. As of the 20th century, groups gained wide recognition by the pioneering work of
Ferdinand Georg Frobenius Ferdinand Georg Frobenius (26 October 1849 – 3 August 1917) was a German mathematician, best known for his contributions to the theory of elliptic functions, differential equations, number theory, and to group theory. He is known for the famous ...
and
William Burnside :''This English mathematician is sometimes confused with the Irish mathematician William S. Burnside (1839–1920).'' __NOTOC__ William Burnside (2 July 1852 – 21 August 1927) was an English mathematician. He is known mostly as an early resea ...
, who worked on
representation theory Representation theory is a branch of mathematics that studies abstract algebraic structures by ''representing'' their elements as linear transformations of vector spaces, and studies modules over these abstract algebraic structures. In esse ...
of finite groups, Richard Brauer's modular representation theory and Issai Schur's papers. The theory of Lie groups, and more generally
locally compact group In mathematics, a locally compact group is a topological group ''G'' for which the underlying topology is locally compact and Hausdorff. Locally compact groups are important because many examples of groups that arise throughout mathematics are loc ...
s was studied by
Hermann Weyl Hermann Klaus Hugo Weyl, (; 9 November 1885 – 8 December 1955) was a German mathematician, theoretical physicist and philosopher. Although much of his working life was spent in Zürich, Switzerland, and then Princeton, New Jersey, he is assoc ...
, Élie Cartan and many others. Its
algebra Algebra () is one of the broad areas of mathematics. Roughly speaking, algebra is the study of mathematical symbols and the rules for manipulating these symbols in formulas; it is a unifying thread of almost all of mathematics. Elementary a ...
ic counterpart, the theory of
algebraic group In mathematics, an algebraic group is an algebraic variety endowed with a group structure which is compatible with its structure as an algebraic variety. Thus the study of algebraic groups belongs both to algebraic geometry and group theory. Man ...
s, was first shaped by Claude Chevalley (from the late 1930s) and later by the work of Armand Borel and
Jacques Tits Jacques Tits () (12 August 1930 – 5 December 2021) was a Belgian-born French mathematician who worked on group theory and incidence geometry. He introduced Tits buildings, the Tits alternative, the Tits group, and the Tits metric. Life ...
. The
University of Chicago The University of Chicago (UChicago, Chicago, U of C, or UChi) is a private research university in Chicago, Illinois. Its main campus is located in Chicago's Hyde Park neighborhood. The University of Chicago is consistently ranked among the b ...
's 1960–61 Group Theory Year brought together group theorists such as Daniel Gorenstein, John G. Thompson and Walter Feit, laying the foundation of a collaboration that, with input from numerous other mathematicians, led to the classification of finite simple groups, with the final step taken by Aschbacher and Smith in 2004. This project exceeded previous mathematical endeavours by its sheer size, in both length of proof and number of researchers. Research concerning this classification proof is ongoing. Group theory remains a highly active mathematical branch, impacting many other fields, as the examples below illustrate.

Elementary consequences of the group axioms

Basic facts about all groups that can be obtained directly from the group axioms are commonly subsumed under ''elementary group theory''. For example, repeated applications of the associativity axiom show that the unambiguity of $a\cdot b\cdot c=(a\cdot b)\cdot c=a\cdot(b\cdot c)$ generalizes to more than three factors. Because this implies that parentheses can be inserted anywhere within such a series of terms, parentheses are usually omitted. Individual axioms may be "weakened" to assert only the existence of a left identity and left inverses. From these ''one-sided axioms'', one can prove that the left identity is also a right identity and a left inverse is also a right inverse for the same element. Since they define exactly the same structures as groups, collectively the axioms are no weaker.

Uniqueness of identity element

The group axioms imply that the identity element is unique: If $e$ and $f$ are identity elements of a group, then $e=e\cdot f=f$. Therefore, it is customary to speak of ''the'' identity.

Uniqueness of inverses

The group axioms also imply that the inverse of each element is unique: If a group element $a$ has both $b$ and $c$ as inverses, then Therefore, it is customary to speak of ''the'' inverse of an element.

Division

Given elements $a$ and $b$ of a group $G$, there is a unique solution $x$ in $G$ to the equation $a\cdot x=b$, namely $a^\cdot b$. (One usually avoids using fraction notation $\tfrac$ unless $G$ is abelian, because of the ambiguity of whether it means $a^\cdot b$ or $b\cdot a^$.) It follows that for each $a$ in $G$, the function $G\to G$ that maps each $x$ to $a\cdot x$ is a
bijection In mathematics, a bijection, also known as a bijective function, one-to-one correspondence, or invertible function, is a function between the elements of two sets, where each element of one set is paired with exactly one element of the other ...
; it is called ''left multiplication by $a$'' or ''left translation by $a$''. Similarly, given $a$ and $b$, the unique solution to $x\cdot a=b$ is $b\cdot a^$. For each $a$, the function $G\to G$ that maps each $x$ to $x\cdot a$ is a bijection called ''right multiplication by $a$'' or ''right translation by $a$''.

Basic concepts

When studying sets, one uses concepts such as
subset In mathematics, set ''A'' is a subset of a set ''B'' if all elements of ''A'' are also elements of ''B''; ''B'' is then a superset of ''A''. It is possible for ''A'' and ''B'' to be equal; if they are unequal, then ''A'' is a proper subset of ...
, function, and quotient by an equivalence relation. When studying groups, one uses instead
subgroup In group theory, a branch of mathematics, given a group ''G'' under a binary operation ∗, a subset ''H'' of ''G'' is called a subgroup of ''G'' if ''H'' also forms a group under the operation ∗. More precisely, ''H'' is a subgro ...
s, homomorphisms, and quotient groups. These are the analogues that take the group structure into account.

Group homomorphisms

Group homomorphisms are functions that respect group structure; they may be used to relate two groups. A ''homomorphism'' from a group $\left(G,\cdot\right)$ to a group $\left(H,*\right)$ is a function $\varphi:G\to H$ such that It would be natural to require also that $\varphi$ respect identities, $\varphi\left(1_G\right)=1_H$, and inverses, $\varphi\left(a^\right)=\varphi\left(a\right)^$ for all $a$ in $G$. However, these additional requirements need not be included in the definition of homomorphisms, because they are already implied by the requirement of respecting the group operation. The ''identity homomorphism'' of a group $G$ is the homomorphism $\iota_G:G\to G$ that maps each element of $G$ to itself. An ''inverse homomorphism'' of a homomorphism $\varphi:G\to H$ is a homomorphism $\psi:H\to G$ such that $\psi\circ\varphi=\iota_G$ and $\varphi\circ\psi=\iota_H$, that is, such that $\psi\bigl\left(\varphi\left(g\right)\bigr\right)=g$ for all $g$ in $G$ and such that $\varphi\bigl\left(\psi\left(h\right)\bigr\right)=h$ for all $h$ in $H$. An ''
isomorphism In mathematics, an isomorphism is a structure-preserving mapping between two structures of the same type that can be reversed by an inverse mapping. Two mathematical structures are isomorphic if an isomorphism exists between them. The word is ...
'' is a homomorphism that has an inverse homomorphism; equivalently, it is a
bijective In mathematics, a bijection, also known as a bijective function, one-to-one correspondence, or invertible function, is a function between the elements of two sets, where each element of one set is paired with exactly one element of the other ...
homomorphism. Groups $G$ and $H$ are called ''isomorphic'' if there exists an isomorphism $\varphi:G\to H$. In this case, $H$ can be obtained from $G$ simply by renaming its elements according to the function $\varphi$; then any statement true for $G$ is true for $H$, provided that any specific elements mentioned in the statement are also renamed. The collection of all groups, together with the homomorphisms between them, form a category, the category of groups.

Subgroups

Informally, a ''subgroup'' is a group $H$ contained within a bigger one, $G$: it has a subset of the elements of $G$, with the same operation. Concretely, this means that the identity element of $G$ must be contained in $H$, and whenever $h_1$ and $h_2$ are both in $H$, then so are $h_1\cdot h_2$ and $h_1^$, so the elements of $H$, equipped with the group operation on $G$ restricted to $H$, indeed form a group. In this case, the inclusion map $H \to G$ is a homomorphism. In the example of symmetries of a square, the identity and the rotations constitute a subgroup $R=\$, highlighted in red in the group table of the example: any two rotations composed are still a rotation, and a rotation can be undone by (i.e., is inverse to) the complementary rotations 270° for 90°, 180° for 180°, and 90° for 270°. The subgroup test provides a necessary and sufficient condition for a nonempty subset ''H'' of a group ''G'' to be a subgroup: it is sufficient to check that $g^\cdot h\in H$ for all elements $g$ and $h$ in $H$. Knowing a group's subgroups is important in understanding the group as a whole. Given any subset $S$ of a group $G$, the subgroup generated by $S$ consists of all products of elements of $S$ and their inverses. It is the smallest subgroup of $G$ containing $S$. In the example of symmetries of a square, the subgroup generated by $r_2$ and $f_$ consists of these two elements, the identity element $\mathrm$, and the element $f_=f_\cdot r_2$. Again, this is a subgroup, because combining any two of these four elements or their inverses (which are, in this particular case, these same elements) yields an element of this subgroup. An injective homomorphism $\phi \colon G\text{'} \to G$ factors canonically as an isomorphism followed by an inclusion, $G\text{'} \;\stackrel\; H \hookrightarrow G$ for some subgroup of . Injective homomorphisms are the
monomorphism In the context of abstract algebra or universal algebra, a monomorphism is an injective homomorphism. A monomorphism from to is often denoted with the notation X\hookrightarrow Y. In the more general setting of category theory, a monomorphism ...
s in the category of groups.

Cosets

In many situations it is desirable to consider two group elements the same if they differ by an element of a given subgroup. For example, in the symmetry group of a square, once any reflection is performed, rotations alone cannot return the square to its original position, so one can think of the reflected positions of the square as all being equivalent to each other, and as inequivalent to the unreflected positions; the rotation operations are irrelevant to the question whether a reflection has been performed. Cosets are used to formalize this insight: a subgroup $H$ determines left and right cosets, which can be thought of as translations of $H$ by an arbitrary group element $g$. In symbolic terms, the ''left'' and ''right'' cosets of $H$, containing an element $g$, are The left cosets of any subgroup $H$ form a partition of $G$; that is, the union of all left cosets is equal to $G$ and two left cosets are either equal or have an empty
intersection In mathematics, the intersection of two or more objects is another object consisting of everything that is contained in all of the objects simultaneously. For example, in Euclidean geometry, when two lines in a plane are not parallel, their ...
. The first case $g_1H=g_2H$ happens precisely when $g_1^\cdot g_2\in H$, i.e., when the two elements differ by an element of $H$. Similar considerations apply to the right cosets of $H$. The left cosets of $H$ may or may not be the same as its right cosets. If they are (that is, if all $g$ in $G$ satisfy $gH=Hg$), then $H$ is said to be a '' normal subgroup''. In $\mathrm_4$, the group of symmetries of a square, with its subgroup $R$ of rotations, the left cosets $gR$ are either equal to $R$, if $g$ is an element of $R$ itself, or otherwise equal to $U=f_R=\$ (highlighted in green in the group table of $\mathrm_4$). The subgroup $R$ is normal, because $f_R=U=Rf_$ and similarly for the other elements of the group. (In fact, in the case of $\mathrm_4$, the cosets generated by reflections are all equal: $f_R=f_R=f_R=f_R$.)

Quotient groups

Suppose that $N$ is a normal subgroup of a group $G$, and $G/N = \$ denotes its set of cosets. Then there is a unique group law on $G/N$ for which the map $G\to G/N$ sending each element $g$ to $gN$ is a homomorphism. Explicitly, the product of two cosets $gN$ and $hN$ is $\left(gh\right)N$, the coset $eN = N$ serves as the identity of $G/N$, and the inverse of $gN$ in the quotient group is . The group $G/N$, read as "$G$ modulo $N$", is called a ''quotient group'' or ''factor group''. The quotient group can alternatively be characterized by a
universal property In mathematics, more specifically in category theory, a universal property is a property that characterizes up to an isomorphism the result of some constructions. Thus, universal properties can be used for defining some objects independently fro ...
. The elements of the quotient group $\mathrm_4/R$ are $R$ and $U=f_R$. The group operation on the quotient is shown in the table. For example, $U\cdot U=f_R\cdot f_R=\left(f_\cdot f_\right)R=R$. Both the subgroup $R=\$ and the quotient $\mathrm_4/R$ are abelian, but $\mathrm_4$ is not. Sometimes a group can be reconstructed from a subgroup and quotient (plus some additional data), by the
semidirect product In mathematics, specifically in group theory, the concept of a semidirect product is a generalization of a direct product. There are two closely related concepts of semidirect product: * an ''inner'' semidirect product is a particular way in ...
construction; $\mathrm_4$ is an example. The first isomorphism theorem implies that any
surjective In mathematics, a surjective function (also known as surjection, or onto function) is a function that every element can be mapped from element so that . In other words, every element of the function's codomain is the image of one element of ...
homomorphism $\phi \colon G \to H$ factors canonically as a quotient homomorphism followed by an isomorphism: $G \to G/\ker \phi \;\stackrel\; H$. Surjective homomorphisms are the
epimorphism In category theory, an epimorphism (also called an epic morphism or, colloquially, an epi) is a morphism ''f'' : ''X'' → ''Y'' that is right-cancellative in the sense that, for all objects ''Z'' and all morphisms , : g_1 \circ f = g_2 \circ f ...
s in the category of groups.

Presentations

Every group is isomorphic to a quotient of a
free group In mathematics, the free group ''F'S'' over a given set ''S'' consists of all words that can be built from members of ''S'', considering two words to be different unless their equality follows from the group axioms (e.g. ''st'' = ''suu''− ...
, in many ways. For example, the dihedral group $\mathrm_4$ is generated by the right rotation $r_1$ and the reflection $f_$ in a vertical line (every element of $\mathrm_4$ is a finite product of copies of these and their inverses). Hence there is a surjective homomorphism from the free group $\langle r,f \rangle$ on two generators to $\mathrm_4$ sending $r$ to $r_1$ and $f$ to $f_1$. Elements in $\ker \phi$ are called ''relations''; examples include $r^4,r^2,\left(r \cdot f\right)^2$. In fact, it turns out that $\ker \phi$ is the smallest normal subgroup of $\langle r,f \rangle$ containing these three elements; in other words, all relations are consequences of these three. The quotient of the free group by this normal subgroup is denoted $\langle r,f \mid r^4=f^2=\left(r\cdot f\right)^2=1 \rangle$. This is called a ''
presentation A presentation conveys information from a speaker to an audience. Presentations are typically demonstrations, introduction, lecture, or speech meant to inform, persuade, inspire, motivate, build goodwill, or present a new idea/product. Presenta ...
'' of $\mathrm_4$ by generators and relations, because the first isomorphism theorem for yields an isomorphism $\langle r,f \mid r^4=f^2=\left(r\cdot f\right)^2=1 \rangle \to \mathrm_4$. A presentation of a group can be used to construct the Cayley graph, a graphical depiction of a discrete group.

Examples and applications

Examples and applications of groups abound. A starting point is the group $\Z$ of integers with addition as group operation, introduced above. If instead of addition multiplication is considered, one obtains multiplicative groups. These groups are predecessors of important constructions in
abstract algebra In mathematics, more specifically algebra, abstract algebra or modern algebra is the study of algebraic structures. Algebraic structures include groups, rings, fields, modules, vector spaces, lattices, and algebras over a field. The term ''a ...
. Groups are also applied in many other mathematical areas. Mathematical objects are often examined by associating groups to them and studying the properties of the corresponding groups. For example,
Henri Poincaré Jules Henri Poincaré ( S: stress final syllable ; 29 April 1854 – 17 July 1912) was a French mathematician, theoretical physicist, engineer, and philosophy of science, philosopher of science. He is often described as a polymath, a ...
founded what is now called algebraic topology by introducing the
fundamental group In the mathematical field of algebraic topology, the fundamental group of a topological space is the group of the equivalence classes under homotopy of the loops contained in the space. It records information about the basic shape, or holes, ...
. By means of this connection, topological properties such as proximity and continuity translate into properties of groups. For example, elements of the fundamental group are represented by loops. The second image shows some loops in a plane minus a point. The blue loop is considered null-homotopic (and thus irrelevant), because it can be continuously shrunk to a point. The presence of the hole prevents the orange loop from being shrunk to a point. The fundamental group of the plane with a point deleted turns out to be infinite cyclic, generated by the orange loop (or any other loop winding once around the hole). This way, the fundamental group detects the hole. In more recent applications, the influence has also been reversed to motivate geometric constructions by a group-theoretical background. In a similar vein,
geometric group theory Geometric group theory is an area in mathematics devoted to the study of finitely generated groups via exploring the connections between algebraic properties of such groups and topological and geometric properties of spaces on which these groups ...
employs geometric concepts, for example in the study of hyperbolic groups. Further branches crucially applying groups include
algebraic geometry Algebraic geometry is a branch of mathematics, classically studying zeros of multivariate polynomials. Modern algebraic geometry is based on the use of abstract algebraic techniques, mainly from commutative algebra, for solving geometrical ...
and number theory. In addition to the above theoretical applications, many practical applications of groups exist. Cryptography relies on the combination of the abstract group theory approach together with
algorithm In mathematics and computer science, an algorithm () is a finite sequence of rigorous instructions, typically used to solve a class of specific problems or to perform a computation. Algorithms are used as specifications for performing ...
ical knowledge obtained in computational group theory, in particular when implemented for finite groups. Applications of group theory are not restricted to mathematics; sciences such as
physics Physics is the natural science that studies matter, its fundamental constituents, its motion and behavior through space and time, and the related entities of energy and force. "Physical science is that department of knowledge which relat ...
,
chemistry Chemistry is the scientific study of the properties and behavior of matter. It is a natural science that covers the elements that make up matter to the compounds made of atoms, molecules and ions: their composition, structure, properties, ...
and computer science benefit from the concept.

Numbers

Many number systems, such as the integers and the rationals, enjoy a naturally given group structure. In some cases, such as with the rationals, both addition and multiplication operations give rise to group structures. Such number systems are predecessors to more general algebraic structures known as
rings Ring may refer to: * Ring (jewellery), a round band, usually made of metal, worn as ornamental jewelry * To make a sound with a bell, and the sound made by a bell :(hence) to initiate a telephone connection Arts, entertainment and media Film and ...
and fields. Further abstract algebraic concepts such as modules,
vector space In mathematics and physics, a vector space (also called a linear space) is a set whose elements, often called ''vectors'', may be added together and multiplied ("scaled") by numbers called '' scalars''. Scalars are often real numbers, but c ...
s and
algebras In mathematics, an algebra over a field (often simply called an algebra) is a vector space equipped with a bilinear product. Thus, an algebra is an algebraic structure consisting of a set together with operations of multiplication and addition ...
also form groups.

Integers

The group of integers $\Z$ under addition, denoted $\left\left(\Z,+\right\right)$, has been described above. The integers, with the operation of multiplication instead of addition, $\left\left(\Z,\cdot\right\right)$ do ''not'' form a group. The associativity and identity axioms are satisfied, but inverses do not exist: for example, $a=2$ is an integer, but the only solution to the equation $a\cdot b=1$ in this case is $b=\tfrac$, which is a rational number, but not an integer. Hence not every element of $\Z$ has a (multiplicative) inverse.

Rationals

The desire for the existence of multiplicative inverses suggests considering
fractions A fraction (from la, fractus, "broken") represents a part of a whole or, more generally, any number of equal parts. When spoken in everyday English, a fraction describes how many parts of a certain size there are, for example, one-half, eight ...
$\frac.$ Fractions of integers (with $b$ nonzero) are known as rational numbers. The set of all such irreducible fractions is commonly denoted $\Q$. There is still a minor obstacle for $\left\left(\Q,\cdot\right\right)$, the rationals with multiplication, being a group: because zero does not have a multiplicative inverse (i.e., there is no $x$ such that $x\cdot 0=1$), $\left\left(\Q,\cdot\right\right)$ is still not a group. However, the set of all ''nonzero'' rational numbers $\Q\smallsetminus\left\=\left\$ does form an abelian group under multiplication, also denoted Associativity and identity element axioms follow from the properties of integers. The closure requirement still holds true after removing zero, because the product of two nonzero rationals is never zero. Finally, the inverse of $a/b$ is $b/a$, therefore the axiom of the inverse element is satisfied. The rational numbers (including zero) also form a group under addition. Intertwining addition and multiplication operations yields more complicated structures called rings and – if
division Division or divider may refer to: Mathematics *Division (mathematics), the inverse of multiplication * Division algorithm, a method for computing the result of mathematical division Military *Division (military), a formation typically consisting ...
by other than zero is possible, such as in $\Q$ – fields, which occupy a central position in abstract algebra. Group theoretic arguments therefore underlie parts of the theory of those entities.

Modular arithmetic

Modular arithmetic for a ''modulus'' $n$ defines any two elements $a$ and $b$ that differ by a multiple of $n$ to be equivalent, denoted by $a \equiv b\pmod$. Every integer is equivalent to one of the integers from $0$ to $n-1$, and the operations of modular arithmetic modify normal arithmetic by replacing the result of any operation by its equivalent
representative Representative may refer to: Politics *Representative democracy, type of democracy in which elected officials represent a group of people *House of Representatives, legislative body in various countries or sub-national entities *Legislator, someon ...
. Modular addition, defined in this way for the integers from $0$ to $n-1$, forms a group, denoted as $\mathrm_n$ or $\left(\Z/n\Z,+\right)$, with $0$ as the identity element and $n-a$ as the inverse element of $a$. A familiar example is addition of hours on the face of a
clock A clock or a timepiece is a device used to measure and indicate time. The clock is one of the oldest human inventions, meeting the need to measure intervals of time shorter than the natural units such as the day, the lunar month and th ...
, where 12 rather than 0 is chosen as the representative of the identity. If the hour hand is on $9$ and is advanced $4$ hours, it ends up on $1$, as shown in the illustration. This is expressed by saying that $9+4$ is congruent to $1$ "modulo $12$" or, in symbols, $9+4\equiv 1 \pmod.$ For any prime number $p$, there is also the multiplicative group of integers modulo $p$. Its elements can be represented by $1$ to $p-1$. The group operation, multiplication modulo $p$, replaces the usual product by its representative, the
remainder In mathematics, the remainder is the amount "left over" after performing some computation. In arithmetic, the remainder is the integer "left over" after dividing one integer by another to produce an integer quotient ( integer division). In alge ...
of division by $p$. For example, for $p=5$, the four group elements can be represented by $1,2,3,4$. In this group, $4\cdot 4\equiv 1\bmod 5$, because the usual product $16$ is equivalent to $1$: when divided by $5$ it yields a remainder of $1$. The primality of $p$ ensures that the usual product of two representatives is not divisible by $p$, and therefore that the modular product is nonzero. The identity element is represented and associativity follows from the corresponding property of the integers. Finally, the inverse element axiom requires that given an integer $a$ not divisible by $p$, there exists an integer $b$ such that $a\cdot b\equiv 1\pmod,$ that is, such that $p$ evenly divides $a\cdot b-1$. The inverse $b$ can be found by using Bézout's identity and the fact that the
greatest common divisor In mathematics, the greatest common divisor (GCD) of two or more integers, which are not all zero, is the largest positive integer that divides each of the integers. For two integers ''x'', ''y'', the greatest common divisor of ''x'' and ''y'' is ...
$\gcd\left(a,p\right)$ In the case $p=5$ above, the inverse of the element represented by $4$ is that represented by $4$, and the inverse of the element represented by $3$ is represented , as $3\cdot 2=6\equiv 1\bmod$. Hence all group axioms are fulfilled. This example is similar to $\left\left(\Q\smallsetminus\left\,\cdot\right\right)$ above: it consists of exactly those elements in the ring $\Z/p\Z$ that have a multiplicative inverse. These groups, denoted $\mathbb F_p^\times$, are crucial to
public-key cryptography Public-key cryptography, or asymmetric cryptography, is the field of cryptographic systems that use pairs of related keys. Each key pair consists of a public key and a corresponding private key. Key pairs are generated with cryptographic a ...
.

Cyclic groups

A ''cyclic group'' is a group all of whose elements are
powers Powers may refer to: Arts and media * ''Powers'' (comics), a comic book series by Brian Michael Bendis and Michael Avon Oeming ** ''Powers'' (American TV series), a 2015–2016 series based on the comics * ''Powers'' (British TV series), a 200 ...
of a particular element $a$. In multiplicative notation, the elements of the group are $\dots, a^, a^, a^, a^0, a, a^2, a^3, \dots,$ where $a^2$ means $a\cdot a$, $a^$ stands for $a^\cdot a^\cdot a^=\left(a\cdot a\cdot a\right)^$, etc. Such an element $a$ is called a generator or a primitive element of the group. In additive notation, the requirement for an element to be primitive is that each element of the group can be written as $\dots, (-a)+(-a), -a, 0, a, a+a, \dots.$ In the groups $\left(\Z/n\Z,+\right)$ introduced above, the element $1$ is primitive, so these groups are cyclic. Indeed, each element is expressible as a sum all of whose terms are $1$. Any cyclic group with $n$ elements is isomorphic to this group. A second example for cyclic groups is the group of $n$th complex roots of unity, given by
complex number In mathematics, a complex number is an element of a number system that extends the real numbers with a specific element denoted , called the imaginary unit and satisfying the equation i^= -1; every complex number can be expressed in the f ...
s $z$ satisfying $z^n=1$. These numbers can be visualized as the vertices on a regular $n$-gon, as shown in blue in the image for $n=6$. The group operation is multiplication of complex numbers. In the picture, multiplying with $z$ corresponds to a counter-clockwise rotation by 60°. From field theory, the group $\mathbb F_p^\times$ is cyclic for prime $p$: for example, if $p=5$, $3$ is a generator since $3^1=3$, $3^2=9\equiv 4$, $3^3\equiv 2$, and $3^4\equiv 1$. Some cyclic groups have an infinite number of elements. In these groups, for every non-zero element $a$, all the powers of $a$ are distinct; despite the name "cyclic group", the powers of the elements do not cycle. An infinite cyclic group is isomorphic to $\left(\Z, +\right)$, the group of integers under addition introduced above. As these two prototypes are both abelian, so are all cyclic groups. The study of finitely generated abelian groups is quite mature, including the
fundamental theorem of finitely generated abelian groups In abstract algebra, an abelian group (G,+) is called finitely generated if there exist finitely many elements x_1,\dots,x_s in G such that every x in G can be written in the form x = n_1x_1 + n_2x_2 + \cdots + n_sx_s for some integers n_1,\dots, n ...
; and reflecting this state of affairs, many group-related notions, such as
center Center or centre may refer to: Mathematics * Center (geometry), the middle of an object * Center (algebra), used in various contexts ** Center (group theory) ** Center (ring theory) * Graph center, the set of all vertices of minimum eccentr ...
and
commutator In mathematics, the commutator gives an indication of the extent to which a certain binary operation fails to be commutative. There are different definitions used in group theory and ring theory. Group theory The commutator of two elements, a ...
, describe the extent to which a given group is not abelian.

Symmetry groups

''Symmetry groups'' are groups consisting of symmetries of given mathematical objects, principally geometric entities, such as the symmetry group of the square given as an introductory example above, although they also arise in algebra such as the symmetries among the roots of polynomial equations dealt with in Galois theory (see below). Conceptually, group theory can be thought of as the study of symmetry. Symmetries in mathematics greatly simplify the study of geometrical or analytical objects. A group is said to act on another mathematical object ''X'' if every group element can be associated to some operation on ''X'' and the composition of these operations follows the group law. For example, an element of the (2,3,7) triangle group acts on a triangular
tiling Tiling may refer to: *The physical act of laying tiles *Tessellations Computing *The compiler optimization of loop tiling * Tiled rendering, the process of subdividing an image by regular grid * Tiling window manager People * Heinrich Sylveste ...
of the hyperbolic plane by permuting the triangles. By a group action, the group pattern is connected to the structure of the object being acted on. In chemical fields, such as
crystallography Crystallography is the experimental science of determining the arrangement of atoms in crystalline solids. Crystallography is a fundamental subject in the fields of materials science and solid-state physics (condensed matter physics). The wor ...
,
space group In mathematics, physics and chemistry, a space group is the symmetry group of an object in space, usually in three dimensions. The elements of a space group (its symmetry operations) are the rigid transformations of an object that leave it unc ...
s and point groups describe molecular symmetries and crystal symmetries. These symmetries underlie the chemical and physical behavior of these systems, and group theory enables simplification of
quantum mechanical Quantum mechanics is a fundamental theory in physics that provides a description of the physical properties of nature at the scale of atoms and subatomic particles. It is the foundation of all quantum physics including quantum chemistry, q ...
analysis of these properties. For example, group theory is used to show that optical transitions between certain quantum levels cannot occur simply because of the symmetry of the states involved. Group theory helps predict the changes in physical properties that occur when a material undergoes a
phase transition In chemistry, thermodynamics, and other related fields, a phase transition (or phase change) is the physical process of transition between one state of a medium and another. Commonly the term is used to refer to changes among the basic states ...
, for example, from a cubic to a tetrahedral crystalline form. An example is
ferroelectric Ferroelectricity is a characteristic of certain materials that have a spontaneous electric polarization that can be reversed by the application of an external electric field. All ferroelectrics are also piezoelectric and pyroelectric, with the ad ...
materials, where the change from a paraelectric to a ferroelectric state occurs at the Curie temperature and is related to a change from the high-symmetry paraelectric state to the lower symmetry ferroelectric state, accompanied by a so-called soft
phonon In physics, a phonon is a collective excitation in a periodic, elastic arrangement of atoms or molecules in condensed matter, specifically in solids and some liquids. A type of quasiparticle, a phonon is an excited state in the quantum mechanica ...
mode, a vibrational lattice mode that goes to zero frequency at the transition. Such spontaneous symmetry breaking has found further application in elementary particle physics, where its occurrence is related to the appearance of Goldstone bosons. Finite symmetry groups such as the Mathieu groups are used in
coding theory Coding theory is the study of the properties of codes and their respective fitness for specific applications. Codes are used for data compression, cryptography, error detection and correction, data transmission and data storage. Codes are studi ...
, which is in turn applied in
error correction In information theory and coding theory with applications in computer science and telecommunication, error detection and correction (EDAC) or error control are techniques that enable reliable delivery of digital data over unreliable communi ...
of transmitted data, and in CD players. Another application is differential Galois theory, which characterizes functions having
antiderivative In calculus, an antiderivative, inverse derivative, primitive function, primitive integral or indefinite integral of a function is a differentiable function whose derivative is equal to the original function . This can be stated symbolic ...
s of a prescribed form, giving group-theoretic criteria for when solutions of certain
differential equation In mathematics, a differential equation is an equation that relates one or more unknown functions and their derivatives. In applications, the functions generally represent physical quantities, the derivatives represent their rates of change, an ...
s are well-behaved. Geometric properties that remain stable under group actions are investigated in (geometric)
invariant theory Invariant theory is a branch of abstract algebra dealing with actions of groups on algebraic varieties, such as vector spaces, from the point of view of their effect on functions. Classically, the theory dealt with the question of explicit descri ...
.

General linear group and representation theory

Matrix groups consist of
matrices Matrix most commonly refers to: * ''The Matrix'' (franchise), an American media franchise ** ''The Matrix'', a 1999 science-fiction action film ** "The Matrix", a fictional setting, a virtual reality environment, within ''The Matrix'' (franchis ...
together with matrix multiplication. The ''general linear group'' $\mathrm \left(n, \R\right)$ consists of all invertible $n$-by-$n$ matrices with real entries. Its subgroups are referred to as ''matrix groups'' or '' linear groups''. The dihedral group example mentioned above can be viewed as a (very small) matrix group. Another important matrix group is the special orthogonal group $\mathrm\left(n\right)$. It describes all possible rotations in $n$ dimensions. Rotation matrices in this group are used in
computer graphics Computer graphics deals with generating images with the aid of computers. Today, computer graphics is a core technology in digital photography, film, video games, cell phone and computer displays, and many specialized applications. A great deal ...
. ''Representation theory'' is both an application of the group concept and important for a deeper understanding of groups. It studies the group by its group actions on other spaces. A broad class of
group representation In the mathematical field of representation theory, group representations describe abstract groups in terms of bijective linear transformations of a vector space to itself (i.e. vector space automorphisms); in particular, they can be used to re ...
s are linear representations in which the group acts on a vector space, such as the three-dimensional
Euclidean space Euclidean space is the fundamental space of geometry, intended to represent physical space. Originally, that is, in Euclid's ''Elements'', it was the three-dimensional space of Euclidean geometry, but in modern mathematics there are Euclidean ...
$\R^3$. A representation of a group $G$ on an $n$-
dimension In physics and mathematics, the dimension of a mathematical space (or object) is informally defined as the minimum number of coordinates needed to specify any point within it. Thus, a line has a dimension of one (1D) because only one coordina ...
al real vector space is simply a group homomorphism $\rho : G \to \mathrm \left(n, \R\right)$ from the group to the general linear group. This way, the group operation, which may be abstractly given, translates to the multiplication of matrices making it accessible to explicit computations. A group action gives further means to study the object being acted on. On the other hand, it also yields information about the group. Group representations are an organizing principle in the theory of finite groups, Lie groups, algebraic groups and
topological group In mathematics, topological groups are logically the combination of groups and topological spaces, i.e. they are groups and topological spaces at the same time, such that the continuity condition for the group operations connects these two s ...
s, especially (locally) compact groups.

Galois groups

''Galois groups'' were developed to help solve polynomial equations by capturing their symmetry features. For example, the solutions of the
quadratic equation In algebra, a quadratic equation () is any equation that can be rearranged in standard form as ax^2 + bx + c = 0\,, where represents an unknown value, and , , and represent known numbers, where . (If and then the equation is linear, not quad ...
$ax^2+bx+c=0$ are given by $x = \frac.$ Each solution can be obtained by replacing the $\pm$ sign by $+$ or $-$; analogous formulae are known for
cubic Cubic may refer to: Science and mathematics * Cube (algebra), "cubic" measurement * Cube, a three-dimensional solid object bounded by six square faces, facets or sides, with three meeting at each vertex ** Cubic crystal system, a crystal system w ...
and quartic equations, but do ''not'' exist in general for degree 5 and higher. In the quadratic formula, changing the sign (permuting the resulting two solutions) can be viewed as a (very simple) group operation. Analogous Galois groups act on the solutions of higher-degree polynomials and are closely related to the existence of formulas for their solution. Abstract properties of these groups (in particular their solvability) give a criterion for the ability to express the solutions of these polynomials using solely addition, multiplication, and
roots A root is the part of a plant, generally underground, that anchors the plant body, and absorbs and stores water and nutrients. Root or roots may also refer to: Art, entertainment, and media * ''The Root'' (magazine), an online magazine focusing ...
similar to the formula above. Modern Galois theory generalizes the above type of Galois groups by shifting to field theory and considering field extensions formed as the splitting field of a polynomial. This theory establishes—via the fundamental theorem of Galois theory—a precise relationship between fields and groups, underlining once again the ubiquity of groups in mathematics.

Finite groups

A group is called ''finite'' if it has a finite number of elements. The number of elements is called the
order Order, ORDER or Orders may refer to: * Categorization, the process in which ideas and objects are recognized, differentiated, and understood * Heterarchy, a system of organization wherein the elements have the potential to be ranked a number of ...
of the group. An important class is the ''
symmetric group In abstract algebra, the symmetric group defined over any set is the group whose elements are all the bijections from the set to itself, and whose group operation is the composition of functions. In particular, the finite symmetric grou ...
s'' $\mathrm_N$, the groups of permutations of $N$ objects. For example, the symmetric group on 3 letters $\mathrm_3$ is the group of all possible reorderings of the objects. The three letters ABC can be reordered into ABC, ACB, BAC, BCA, CAB, CBA, forming in total 6 ( factorial of 3) elements. The group operation is composition of these reorderings, and the identity element is the reordering operation that leaves the order unchanged. This class is fundamental insofar as any finite group can be expressed as a subgroup of a symmetric group $\mathrm_N$ for a suitable integer $N$, according to
Cayley's theorem In group theory, Cayley's theorem, named in honour of Arthur Cayley, states that every group is isomorphic to a subgroup of a symmetric group. More specifically, is isomorphic to a subgroup of the symmetric group \operatorname(G) whose eleme ...
. Parallel to the group of symmetries of the square above, $\mathrm_3$ can also be interpreted as the group of symmetries of an
equilateral triangle In geometry, an equilateral triangle is a triangle in which all three sides have the same length. In the familiar Euclidean geometry, an equilateral triangle is also equiangular; that is, all three internal angles are also congruent to each oth ...
. The order of an element $a$ in a group $G$ is the least positive integer $n$ such that $a^n=e$, where $a^n$ represents $\underbrace_,$ that is, application of the operation "$\cdot$" to $n$ copies of $a$. (If "$\cdot$" represents multiplication, then $a^n$ corresponds to the $n$th power of $a$.) In infinite groups, such an $n$ may not exist, in which case the order of $a$ is said to be infinity. The order of an element equals the order of the cyclic subgroup generated by this element. More sophisticated counting techniques, for example, counting cosets, yield more precise statements about finite groups: Lagrange's Theorem states that for a finite group $G$ the order of any finite subgroup $H$
divides In mathematics, a divisor of an integer n, also called a factor of n, is an integer m that may be multiplied by some integer to produce n. In this case, one also says that n is a multiple of m. An integer n is divisible or evenly divisible by ...
the order of $G$. The
Sylow theorems In mathematics, specifically in the field of finite group theory, the Sylow theorems are a collection of theorems named after the Norwegian mathematician Peter Ludwig Sylow that give detailed information about the number of subgroups of fix ...
give a partial converse. The dihedral group $\mathrm_4$ of symmetries of a square is a finite group of order 8. In this group, the order of $r_1$ is 4, as is the order of the subgroup $R$ that this element generates. The order of the reflection elements $f_$ etc. is 2. Both orders divide 8, as predicted by Lagrange's theorem. The groups $\mathbb F_p^\times$ of multiplication modulo a prime $p$ have order $p-1$.

Finite abelian groups

Any finite abelian group is isomorphic to a
product Product may refer to: Business * Product (business), an item that serves as a solution to a specific consumer problem. * Product (project management), a deliverable or set of deliverables that contribute to a business solution Mathematics * Pr ...
of finite cyclic groups; this statement is part of the
fundamental theorem of finitely generated abelian groups In abstract algebra, an abelian group (G,+) is called finitely generated if there exist finitely many elements x_1,\dots,x_s in G such that every x in G can be written in the form x = n_1x_1 + n_2x_2 + \cdots + n_sx_s for some integers n_1,\dots, n ...
. Any group of prime order $p$ is isomorphic to the cyclic group $\mathrm_p$ (a consequence of Lagrange's theorem). Any group of order $p^2$ is abelian, isomorphic to $\mathrm_$ or $\mathrm_p \times \mathrm_p$. But there exist nonabelian groups of order $p^3$; the dihedral group $\mathrm_4$ of order $2^3$ above is an example.

Simple groups

When a group $G$ has a normal subgroup $N$ other than $\$ and $G$ itself, questions about $G$ can sometimes be reduced to questions about $N$ and $G/N$. A nontrivial group is called ''
simple Simple or SIMPLE may refer to: *Simplicity, the state or quality of being simple Arts and entertainment * ''Simple'' (album), by Andy Yorke, 2008, and its title track * "Simple" (Florida Georgia Line song), 2018 * "Simple", a song by Johnn ...
'' if it has no such normal subgroup. Finite simple groups are to finite groups as prime numbers are to positive integers: they serve as building blocks, in a sense made precise by the Jordan–Hölder theorem.

Classification of finite simple groups

Computer algebra system A computer algebra system (CAS) or symbolic algebra system (SAS) is any mathematical software with the ability to manipulate mathematical expressions in a way similar to the traditional manual computations of mathematicians and scientists. Th ...
s have been used to list all groups of order up to 2000. But classifying all finite groups is a problem considered too hard to be solved. The classification of all finite ''simple'' groups was a major achievement in contemporary group theory. There are several infinite families of such groups, as well as 26 "
sporadic groups In mathematics, a sporadic group is one of the 26 exceptional Group (mathematics), groups found in the classification of finite simple groups. A simple group is a group ''G'' that does not have any normal subgroups except for the trivial group a ...
" that do not belong to any of the families. The largest sporadic group is called the monster group. The
monstrous moonshine In mathematics, monstrous moonshine, or moonshine theory, is the unexpected connection between the monster group ''M'' and modular functions, in particular, the ''j'' function. The term was coined by John Conway and Simon P. Norton in 1979 ...
conjectures, proved by Richard Borcherds, relate the monster group to certain modular functions. The gap between the classification of simple groups and the classification of all groups lies in the extension problem..

An equivalent definition of group consists of replacing the "there exist" part of the group axioms by operations whose result is the element that must exist. So, a group is a set $G$ equipped with a binary operation $G \times G \rightarrow G$ (the group operation), a
unary operation In mathematics, an unary operation is an operation with only one operand, i.e. a single input. This is in contrast to binary operations, which use two operands. An example is any function , where is a set. The function is a unary operation o ...
$G \rightarrow G$ (which provides the inverse) and a nullary operation, which has no operand and results in the identity element. Otherwise, the group axioms are exactly the same. This variant of the definition avoids existential quantifiers and is used in computing with groups and for computer-aided proofs. This way of defining groups lends itself to generalizations such as the notion of
group object In category theory, a branch of mathematics, group objects are certain generalizations of groups that are built on more complicated structures than sets. A typical example of a group object is a topological group, a group whose underlying set is ...
in a category. Briefly, this is an object with
morphism In mathematics, particularly in category theory, a morphism is a structure-preserving map from one mathematical structure to another one of the same type. The notion of morphism recurs in much of contemporary mathematics. In set theory, morphisms ...
s that mimic the group axioms.

Topological groups

Some
topological space In mathematics, a topological space is, roughly speaking, a geometrical space in which closeness is defined but cannot necessarily be measured by a numeric distance. More specifically, a topological space is a set whose elements are called points ...
s may be endowed with a group law. In order for the group law and the topology to interweave well, the group operations must be continuous functions; informally, $g \cdot h$ and $g^$ must not vary wildly if $g$ and $h$ vary only a little. Such groups are called ''topological groups,'' and they are the group objects in the category of topological spaces. The most basic examples are the group of real numbers under addition and the group of nonzero real numbers under multiplication. Similar examples can be formed from any other
topological field In mathematics, a field is a set on which addition, subtraction, multiplication, and division are defined and behave as the corresponding operations on rational and real numbers do. A field is thus a fundamental algebraic structure which is wi ...
, such as the field of complex numbers or the field of -adic numbers. These examples are
locally compact In topology and related branches of mathematics, a topological space is called locally compact if, roughly speaking, each small portion of the space looks like a small portion of a compact space. More precisely, it is a topological space in which ev ...
, so they have
Haar measure In mathematical analysis, the Haar measure assigns an "invariant volume" to subsets of locally compact topological groups, consequently defining an integral for functions on those groups. This measure was introduced by Alfréd Haar in 1933, thoug ...
s and can be studied via
harmonic analysis Harmonic analysis is a branch of mathematics concerned with the representation of functions or signals as the superposition of basic waves, and the study of and generalization of the notions of Fourier series and Fourier transforms (i.e. an ex ...
. Other locally compact topological groups include the group of points of an algebraic group over a local field or adele ring; these are basic to number theory Galois groups of infinite algebraic field extensions are equipped with the Krull topology, which plays a role in infinite Galois theory. A generalization used in algebraic geometry is the étale fundamental group.

Lie groups

A ''Lie group'' is a group that also has the structure of a
differentiable manifold In mathematics, a differentiable manifold (also differential manifold) is a type of manifold that is locally similar enough to a vector space to allow one to apply calculus. Any manifold can be described by a collection of charts (atlas). One ma ...
; informally, this means that it looks locally like a Euclidean space of some fixed dimension. Again, the definition requires the additional structure, here the manifold structure, to be compatible: the multiplication and inverse maps are required to be
smooth Smooth may refer to: Mathematics * Smooth function, a function that is infinitely differentiable; used in calculus and topology * Smooth manifold, a differentiable manifold for which all the transition maps are smooth functions * Smooth algebraic ...
. A standard example is the general linear group introduced above: it is an open subset of the space of all $n$-by-$n$ matrices, because it is given by the inequality $\det (A) \ne 0,$ where $A$ denotes an $n$-by-$n$ matrix. Lie groups are of fundamental importance in modern physics:
Noether's theorem Noether's theorem or Noether's first theorem states that every differentiable symmetry of the action of a physical system with conservative forces has a corresponding conservation law. The theorem was proven by mathematician Emmy Noether ...
links continuous symmetries to conserved quantities.
Rotation Rotation, or spin, is the circular movement of an object around a '' central axis''. A two-dimensional rotating object has only one possible central axis and can rotate in either a clockwise or counterclockwise direction. A three-dimensional ...
, as well as translations in
space Space is the boundless three-dimensional extent in which objects and events have relative position and direction. In classical physics, physical space is often conceived in three linear dimensions, although modern physicists usually consider ...
and time, are basic symmetries of the laws of
mechanics Mechanics (from Ancient Greek: μηχανική, ''mēkhanikḗ'', "of machines") is the area of mathematics and physics concerned with the relationships between force, matter, and motion among physical objects. Forces applied to objects re ...
. They can, for instance, be used to construct simple models—imposing, say, axial symmetry on a situation will typically lead to significant simplification in the equations one needs to solve to provide a physical description. Another example is the group of Lorentz transformations, which relate measurements of time and velocity of two observers in motion relative to each other. They can be deduced in a purely group-theoretical way, by expressing the transformations as a rotational symmetry of
Minkowski space In mathematical physics, Minkowski space (or Minkowski spacetime) () is a combination of three-dimensional Euclidean space and time into a four-dimensional manifold where the spacetime interval between any two events is independent of the inerti ...
. The latter serves—in the absence of significant
gravitation In physics, gravity () is a fundamental interaction which causes mutual attraction between all things with mass or energy. Gravity is, by far, the weakest of the four fundamental interactions, approximately 1038 times weaker than the strong ...
—as a model of
spacetime In physics, spacetime is a mathematical model that combines the three dimensions of space and one dimension of time into a single four-dimensional manifold. Spacetime diagrams can be used to visualize relativistic effects, such as why differe ...
in
special relativity In physics, the special theory of relativity, or special relativity for short, is a scientific theory regarding the relationship between space and time. In Albert Einstein's original treatment, the theory is based on two postulates: # The laws ...
. The full symmetry group of Minkowski space, i.e., including translations, is known as the Poincaré group. By the above, it plays a pivotal role in special relativity and, by implication, for quantum field theories. Symmetries that vary with location are central to the modern description of physical interactions with the help of
gauge theory In physics, a gauge theory is a type of field theory in which the Lagrangian (and hence the dynamics of the system itself) does not change (is invariant) under local transformations according to certain smooth families of operations ( Lie grou ...
. An important example of a gauge theory is the
Standard Model The Standard Model of particle physics is the theory describing three of the four known fundamental forces (electromagnetic, weak and strong interactions - excluding gravity) in the universe and classifying all known elementary particles. It ...
, which describes three of the four known
fundamental force In physics, the fundamental interactions, also known as fundamental forces, are the interactions that do not appear to be reducible to more basic interactions. There are four fundamental interactions known to exist: the gravitational and electrom ...
s and classifies all known
elementary particle In particle physics, an elementary particle or fundamental particle is a subatomic particle that is not composed of other particles. Particles currently thought to be elementary include electrons, the fundamental fermions ( quarks, leptons, ant ...
s.

Generalizations

More general structures may be defined by relaxing some of the axioms defining a group. The table gives a list of several structures generalizing groups. For example, if the requirement that every element has an inverse is eliminated, the resulting algebraic structure is called a monoid. The natural numbers $\mathbb N$ (including zero) under addition form a monoid, as do the nonzero integers under multiplication $\left(\Z \smallsetminus \, \cdot\right)$. Adjoining inverses of all elements of the monoid $\left(\Z \smallsetminus \, \cdot\right)$ produces a group $\left(\Q \smallsetminus \, \cdot\right)$, and likewise adjoining inverses to any (abelian) monoid produces a group known as the Grothendieck group of . A group can be thought of as a
small category In mathematics, a category (sometimes called an abstract category to distinguish it from a concrete category) is a collection of "objects" that are linked by "arrows". A category has two basic properties: the ability to compose the arrows ass ...
with one object in which every morphism is an isomorphism: given such a category, the set $\operatorname\left(x,x\right)$ is a group; conversely, given a group , one can build a small category with one object in which $\operatorname\left(x,x\right) \simeq G$. More generally, a
groupoid In mathematics, especially in category theory and homotopy theory, a groupoid (less often Brandt groupoid or virtual group) generalises the notion of group in several equivalent ways. A groupoid can be seen as a: *''Group'' with a partial functi ...
is any small category in which every morphism is an isomorphism. In a groupoid, the set of all morphisms in the category is usually not a group, because the composition is only partially defined: is defined only when the source of matches the target of . Groupoids arise in topology (for instance, the fundamental groupoid) and in the theory of stacks. Finally, it is possible to generalize any of these concepts by replacing the binary operation with an -ary operation (i.e., an operation taking arguments, for some nonnegative integer ). With the proper generalization of the group axioms, this gives a notion of -ary group.

* List of group theory topics

References

General references

* , Chapter 2 contains an undergraduate-level exposition of the notions covered in this article. * * , an elementary introduction. * . * . * * . * . * . * .

Special references

* . * . * * * . * . * . * . * . * * . * . * . * . * . * * . * . * * . * * . * * . * . * . * . * . * . * * * . * * . * . * * . * . * . * . * . * . * . * * * * * . * . * * . * . * . *

Historical references

* * . * * . * . * (Galois work was first published by Joseph Liouville in 1843). * . * . * . * * . * . * .